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(12.6.5)   Let 3? be a subset of the set of continuous mappings of E into R, with
the following three properties:

(1)  for each pair of disjoint nonempty subsets M, N of E, with M compact
and N closed, there exists a function/^ 0 in J^ which is equal to 0 on N and
^1 onM;

(2)  if (/a)a 61 is a family of functions in 3F whose supports form a locally
finite family, then  fa belongs to ^;

ae I

(3)  if/e & is such that/(jc) > 0 for all x e E, then g/fe & for all g e &.

Then the conclusions of (12.6.3) and (12.6.4) are still valid, when the An
are relatively compact, if we impose the extra condition that the functions/,
should belong to !F. The proofs are unaltered.


In this section, and above all in Chapter XIII, we shall need to consider
mappings of a set A into the extended real line R. Such mappings we shall call
(by abuse of language) real-valued functions on A. Such a function/is said to
be finite if its value at each point a e A is finite, i.e., if/(A) c R. The function/
is said to be majorized or bounded above (resp. minorized or bounded below)
if there exists a finite majorant (resp. a finite minorant) of/(A). If/is both
majorized and minorized, it is said to be bounded (which clearly implies that
/is finite).

We recall ((4.1.8), (4.1.9), and (3.15.5)) that in R x I the function
(x, y)\-^x + y (resp. (x, y) H* xy) is defined and continuous except at the points
(+00, -oo) and (-co, +00) (resp. (+00, 0), (-oo,0), (0, +00), (0, -oo)).
The function xt-+l/x on E is defined and continuous except at the point
x = 0. In the interval [0, +00] of K, the function x\-*l/x (defined on
]0, +00]) can be extended by continuity by giving it the value +00 at
the point 0.

Let E be a topological space and let/be a mapping of E into the extended
real line K. The function/is said to be lower (resp. upper) semicontinuous at a
point ;c0 e E if, for each a e K such that a <f(xQ) (resp. a >f(xQ))9 there
exists a neighborhood V of #0 in E such that, for all x e V, we have a <f(x)
(resp. a >/(#)). The function/is said to be lower (resp. upper) semicontinuous
on E if it is so at every point x0 e E.